Different sources of error in Finite Element computations

Question

Consider the problem $-\Delta u = f$ in $\Omega$, with $u=0$ on $\partial \Omega$. Suppose that $\Omega$ is a polygon and that we approximate the solutions to the previous problem using Lagrange finite elements on a triangular mesh. When solving such a finite element problem there are multiple sources of error:

1. The discrete mesh may not be an exact mesh for the polygon due to rounding errors (typically of the order of the machine $\varepsilon$). This gives slight perturbations for the mass and rigidity matrices.

2. The discrete solution comes from solving a linear system. This linear system is solved numerically, so there is an additional error here.

3. The difference between the analytical solution and the finite element one is usually quantified with a priori estimates depending on the mesh size $h$.

Usually only point 3. above is mentioned when dealing with error estimates from finite elements. I guess this is due to the fact that errors coming from points 1 and 2 above (rounding errors, linear systems) are usually orders of magnitude smaller than the discretization error in 3. However, I am interested in knowing more about how one can quantify the errors coming from the first two points and eventual references treating these aspects.

Can you indicate references in the literature or the main ideas that can be used in order to better understand errors coming from points 1-2 above?

Answer 1

The triangle inequality is your friend. Let's ignore the issue of boundary approximation for a moment, then you are computing a solution with inexact linear solver. Let's call it $u_h$. We will call the exact solution of the PDE $u$, and the exact finite element solution $u_\text{FEM}$; neither of these can be computed exactly (in the case of $u_\text{FEM}$ because we can not solve linear systems exactly, but accrue round-off and iteration error).

Then the total error satisfies $$ \|u_h - u\| = \|u_h-u_\text{FEM} + u_\text{FEM} - u\| \le \|u_h-u_\text{FEM}\| + \|u_\text{FEM} - u\|. $$ The first of the terms on the right results from inexact solution of the linear system, and we know how to estimate it. The second of the terms on the right results from finite element approximation (without having to take into account any other approximation), and again we know how to estimate this error.

If you had other error terms -- say, quadrature error or boundary approximation errors -- then you would simply have to add and subtract additional terms in the norm as I've done above with $u_\text{FEM}$ and expand the norm using the triangle inequality into more separate norms that measure exactly one error effect at a time. You don't have to consider the combined effects of all errors at once.

Answer 2

I think that errors 1 and 2 classify as variational crimes, as also classifies approximating the domain by a mesh (but that's not your case). That being said, I am not aware of how big these errors are in comparison with the approximation error.

Now, regarding rounding errors I remember Nick's Trefethen quote [1]

If rounding errors vanished, numerical analysis would remain.

References

1. Trefethen, L. N. (1992). The definition of numerical analysis. Cornell University.

2. Strang, G. (1972). Variational crimes in the finite element method. In The mathematical foundations of the finite element method with applications to partial differential equations (pp. 689-710). Academic Press.

3. Brenner, S. C., Scott, L. R., & Scott, L. R. (2008). Chapter 10: The mathematical theory of finite element methods (Vol. 3). New York: Springer.

4. Holst, M., & Stern, A. (2012). Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. Foundations of Computational Mathematics, 12(3), 263-293.